Real-Time Neuromuscular and Metabolic Fatigue Classification in Sprint and Jump Athletes: An Entropy-Informed Computational Framework for Edge Inference

Abstract

Real-time fatigue classification on resource-constrained edge devices faces three unresolved computational challenges: just-in-time compilation latency spikes that violate the 50 ms inference budget, statistical moment features insensitive to temporal complexity signatures of fatigue, and binary anomaly outputs insufficient for actionable coaching decisions. A synthetic IMU dataset (9 subjects, 540,000 samples, 6 channels at 100 Hz) was generated as a reproducible computational benchmark, with fatigue signatures calibrated to published biomechanical effect sizes (sample entropy $d=+0.77$; permutation entropy $d=+0.38$). We present Safari (Stochastic Adaptive Fitness-Aware Real-time Inference), an end-to-end computational pipeline integrating: a dual-pathway entropy triplet (SampEn, PermEn, SpEn) replacing statistical moments; 16 pre-compiled polyhedral anchor kernels eliminating JIT latency; $O\left({(\Delta W)}^2\right)$-bounded runtime interpolation; subject-specific MaxEnt free-energy anomaly scoring; and a Banister fitness–fatigue adaptive threshold. Safari achieves AUC-ROC = 0.9820 (Monte Carlo 95% CI: 0.9726–0.9886), F1 = 0.8835, four-state accuracy = 83.3%, and worst-case latency = 7.2 ms on a Raspberry Pi 4. Entropy features achieve 1.55× higher discriminability than statistical moments. Safari is a computational framework for real-time fatigue monitoring, contributing a reproducible algorithmic benchmark for edge AI in movement analysis, with real-athlete validation as the recommended next step.

1. Introduction

Real-time fatigue monitoring on edge-deployed wearable devices presents three unresolved computational challenges. First, deterministic execution: inference pipelines on resource-constrained ARM processors must process variable-length stride windows within a 50 ms budget, yet just-in-time (JIT) compiler recompilation for each new window shape introduces latency spikes of 20–55 ms, making worst-case execution time non-deterministic [1,2,3]. Second, algorithmic discriminability: prevailing real-time pipelines extract statistical moments (mean, variance, skewness, kurtosis) that are computationally cheap but informationally insufficient, capturing distributional properties while remaining blind to the temporal complexity changes that characterise both neuromuscular and metabolic fatigue [4,5]. Third, actionable output design: binary anomaly classifiers produce outputs too coarse for real-time coaching decisions; a graduated, multi-state output aligned with established training adaptation theory is required [6,7].

Inertial measurement units (IMUs) are the signal source for the inference pipeline developed in this work. Neuromuscular and metabolic fatigue produce detectable changes in IMU signal complexity before the athlete subjectively experiences performance degradation [4], creating a pre-symptomatic monitoring window. The physiological and biomechanical chain linking fatigue to IMU signal complexity is detailed in Section 3; here we note only that stride-to-stride phase jitter, spectral compression, and amplitude envelope changes are the three measurable manifestations that motivate the entropy feature design at the core of Safari.

Existing systems address at most one of the three challenges in isolation: polyhedral compilation frameworks target deterministic execution but do not address feature design; entropy-based wearable systems improve discriminability but assume fixed window lengths and provide binary outputs; Banister-based coaching tools provide graduated outputs but operate offline, not in real time. No existing system unifies all three capabilities within a single edge-deployable pipeline.

We present Safari (Stochastic Adaptive Fitness-Aware Real-time Inference), an end-to-end computational pipeline (Figure 1) integrating: a dual-pathway entropy triplet for neuromuscular and metabolic fatigue discrimination; polyhedral kernel compilation for deterministic edge inference; and Banister training adaptation theory for session-adaptive threshold evolution. The main computational contributions are (detailed in Section 3 and Section 4):

• Dual-pathway entropy feature triplet. We replace conventional statistical moments with a triplet of physiologically grounded entropy descriptors. $\mathrm{SampEn}$ and $\mathrm{PermEn}$ serve as neuromuscular complexity descriptors, capturing stride irregularity and ordinal pattern breakdown that arise as central motor control degrades. $\mathrm{SpEn}$ serves as the metabolic complexity descriptor, capturing the spectral power compression toward lower frequencies that accompanies fast-twitch motor unit dropout under glycolytic depletion. Together these three measures form a compact, interpretable feature vector that reflects both fatigue pathways simultaneously.
• Parametric polyhedral modelling and shape-aware kernel interpolation. We model the space of admissible stride window lengths as a parametric polyhedron and discretise it into a finite family of anchor points. For each anchor, we apply polyhedral compilation via the Polyhedral Extraction Tool (PET) and Integer Set Library (ISL) to generate a shape-specialised kernel that exploits loop fusion, tiling, and SIMD vectorisation for maximal cache efficiency on the ARM Cortex-A72 processor. At runtime, the engine selects the two nearest anchor kernels, executes them sequentially on the incoming stride window, and synthesises the final entropy feature vector via linear interpolation of their outputs, eliminating just-in-time compilation entirely and guaranteeing a deterministic, bounded worst-case execution time.
• Runtime feature synthesis via interpolation of computational results. The key insight driving the interpolation is that entropy feature vectors are smooth, twice-differentiable functions of the stride window length under mild stationarity conditions. This continuity means that the feature vector for any intermediate window length can be accurately approximated by a convex blend of the vectors produced by the two bracketing anchor kernels, with a blending coefficient proportional to the distance from the lower anchor. The interpolation is element-wise, requires only $3D$ multiply-add operations, and introduces a bounded approximation error that decays as $O\left({(\Delta W)}^2\right)$ with anchor spacing. A two-slot LRU kernel cache further reduces disk access overhead during consecutive strides with similar lengths.
• MaxEnt free-energy anomaly scoring with mixed-effects personalisation. Anomaly scoring is grounded in the maximum entropy principle: a subject-specific one-class support vector machine is trained exclusively on each athlete’s unfatigued baseline windows, and deviations are scored as the free energy $s=-g\left(\tilde{\mathbf{f}}\right)$ under this personal manifold. A random-intercept standardisation applied to the feature vectors prior to classification absorbs the strong inter-athlete entropy baseline variability confirmed by data diagnostics, bringing the effective model closer to the subject-specific paradigm that the literature identifies as essential for accurate fatigue classification.
• Banister fitness–fatigue adaptive threshold. The detection threshold is coupled to the Banister impulse-response model [6], evolving dynamically within each session as the balance between fitness and fatigue components shifts. As the within-session fatigue surplus grows, the threshold tightens, making the system progressively more sensitive, embedding established training adaptation theory directly into the real-time inference pipeline.
• Four-state neuromuscular and metabolic fatigue classification. The continuous free-energy score is mapped to four operational simulation states, Fresh, Accumulating, Fatigued, Critical, using within-session score quartiles. Each state carries a direct biomechanical and coaching interpretation consistent with the published literature [4,7,8], designed, once validated on real athletes, to support proactive training adaptation decisions.

2. Related Work

2.1. AI and Wearable IMU Systems for Fatigue Detection

AI-driven wearable IMU systems have demonstrated strong potential for athlete fatigue monitoring [9,10]. AI has also transformed clinical athlete screening, including cardiovascular pre-participation examination using deep learning [11], confirming the broad impact of AI across athlete health management. Support vector machines applied to sprint IMU data detect neuromuscular fatigue with 87% sensitivity [9]. LSTM networks identify biomechanical deviations up to 2.5 training sessions before symptom onset [7]. The critical role of personalisation is established: subject-specific models achieve 97.7% accuracy (AUC 0.997) versus 55% for population models [4,7]. A scoping review of assessment methods for sport-induced neuromuscular fatigue identified the absence of real-time, objective, personalised monitoring as the primary clinical gap [12]. The influence of fatigue on biomechanical parameters in endurance running has been systematically reviewed, confirming measurable kinematic deterioration including stride frequency reduction, increased ground contact time, and altered trunk kinematics [13]. Eckart et al. [9] identify the absence of real-time, entropy-based, personalised inference pipelines as the primary evidence gap that Safari addresses. Deep learning combining CNN-LSTM-Attention detects fatigue from multimodal sEMG and IMU signals [14]. Runner fatigue stages have been identified from inertial sensors using deep learning [15]. Machine learning for IMU-based upper-extremity exercise classification has demonstrated broad movement generalisability [16]. Sprint fatigue monitoring using countermovement jump force–time profiles has confirmed the utility of objective neuromuscular assessment [17].

2.2. Neuromuscular and Metabolic Fatigue Signatures in IMU Data

Neuromuscular fatigue manifests in IMU signals as increased stride-to-stride variability, reduced autocorrelation, and breakdown of ordinal temporal patterns, captured by rising $\mathrm{SampEn}$ and $\mathrm{PermEn}$ [4,18,19]. Metabolic fatigue drives a distinct signature: spectral power compression toward lower frequencies as fast-twitch motor unit dropout reduces mechanical bandwidth, captured by falling $\mathrm{SpEn}$ [5,20]. Biró et al. [4] demonstrated large effect sizes ($d>1.0$) for entropy features where moments yield non-significant differences. Dimmick et al. [7] showed that PCA of jump force–time profiles can separately identify neuromuscular and metabolic fatigue at different post-exercise time points: metabolic fatigue peaks at 6 h, neuromuscular fatigue persists to 48 h. These distinct time constants motivate the dual-pathway framing of the Safari entropy triplet. No prior work integrates all three entropy descriptors into a polyhedral compilation pipeline for real-time wearable inference.

2.3. Fitness-Fatigue Modelling and Training Adaptation

The Banister impulse-response model [6] underpins training load management by modelling performance as the difference between fitness and fatigue components, each driven by training load with distinct time constants. Extensions using Bayesian estimation [21] and optimal control [22] have refined its predictive accuracy. Safari adopts the impulse-response structure for adaptive thresholding, embedding training adaptation theory directly into real-time detection. The Special Issue theme of AI-driven training adaptation analysis is served by this connection: the four fatigue states and the Banister threshold together provide a session-level picture of how adaptation is progressing.

2.4. Polyhedral Compilation for Dynamic-Shape Inference

TVM [1] and XLA [23] optimise deep-learning kernels for fixed shapes but require JIT recompilation for shape changes. Parametric tiling [24] and multi-versioning [25] address dynamic shapes with efficiency or memory penalties. Configurable polyhedral scheduling [26] is the closest prior work; Safari differs by interpolating entropy feature vector outputs of pre-compiled shape-specialised kernels rather than parameterising the kernel itself, enabling full efficiency without conservative dependency assumptions.

Synthesis: the evidence gap. The reviewed literature establishes that entropy features outperform statistical moments for neuromuscular fatigue detection, that subject-specific models are necessary for clinical accuracy, and that Banister-based threshold adaptation is theoretically well-grounded. What is absent from the existing literature is a unified system that combines all three capabilities: entropy-based feature design, personalised anomaly scoring, and session-adaptive thresholding; within a deterministic, real-time edge inference architecture. Safari addresses precisely this gap.

3. Background and Problem Formulation

3.1. Biomechanical and Metabolic Fatigue in Sprint and Jump Athletes

Let $\mathbf{x}\left(t\right)\in {\mathbb{R}}^D$ denote the IMU signal at time step t, with $D=6$ channels (tri-axial accelerometer and gyroscope) at $f_s=100$ Hz. During a session the athlete completes movement phases indexed $n=1,\dots ,N$ with window lengths

$$W_n\in \mathcal{W}=\{50,51,\dots ,200\}\mathrm{samples},$$

(1)

corresponding to 0.5–2.0 s at 100 Hz.

The lower bound $W_{\min }=50$ samples (0.5 s) corresponds to the minimum ground-contact phase duration observed in jump biomechanics [4]; the upper bound $W_{\max }=200$ samples (2.0 s) covers the maximum walking stride cycle duration relevant to low-intensity recovery intervals [5]. This range was chosen to encompass all locomotion phases encountered in sprint and jump athlete monitoring without requiring stride segmentation to operate outside its validated operating range.

Two distinct fatigue pathways produce measurable complexity changes in the IMU signal:

Neuromuscular pathway [4,5]:

• Stride-to-stride phase jitter accumulates as motor timing deteriorates $\to$$\mathrm{SampEn}$ rises.
• Bilateral amplitude asymmetry increases as left-right coordination degrades $\to$$\mathrm{PermEn}$ rises (ordinal patterns break down).

Metabolic pathway [5,7]:

• Fast-twitch motor unit dropout reduces high-frequency force production $\to$ spectral power compresses toward low frequencies $\to$$\mathrm{SpEn}$ falls.
• Movement efficiency declines (ODBA increases as biomechanical economy degrades) [27].

3.2. The Biomechanical Chain: From Muscle Physiology to IMU Signal Complexity

The rationale for entropy-based features is grounded in a four-level causal chain from exercise physiology to wearable sensor output.

Level 1: Muscle physiology. During repeated maximal sprint and jump efforts, fast-twitch fibre glycogen depletion, lactate accumulation, and rising intramuscular phosphate impair cross-bridge cycling kinetics, reducing peak force and rate of force development [8]. Simultaneously, central fatigue (manifested as a progressive reduction in voluntary activation) further limits motor unit discharge rates.

Level 2: Neuromuscular coordination. Reduced motor unit firing rates disrupt the finely timed inter-muscular coordination patterns that characterise efficient sprint and jump mechanics. Bilateral symmetry deteriorates as the more-fatigued limb adopts compensatory motor strategies, and the stretch-shortening cycle efficiency of the muscle-tendon unit declines as tendon elastic recoil is impaired [28]. A critical aspect of training is the assessment of internal load, specifically athletes’ psychophysiological response to training, using both subjective and objective measurements, crucial for enhancing performance and preventing training-related injuries [9].

Level 3: Biomechanics and kinematics. The neuromuscular changes propagate into observable kinematic alterations: stride length shortens, ground contact time increases, trunk sway amplitude rises, knee flexion at initial contact decreases, and bilateral ground contact asymmetry grows. These are the variables traditionally assessed via force plates, motion capture, and video analysis [4]. The four fatigue states in Safari correspond to escalating severity along this kinematic deterioration continuum: Fresh (normal kinematics), Accumulating (subtle symmetry loss), Fatigued (measurable stride irregularity and contact time increase), and Critical (injury-risk compensatory patterns).

Level 4: IMU signal complexity. Kinematic changes at Level 3 alter the temporal structure of the lumbar-mounted IMU signal in three measurable ways. First, stride-to-stride phase jitter increases as motor unit timing variability grows, raising $\mathrm{SampEn}$ (temporal irregularity). Second, bilateral asymmetry disrupts the ordinal sequence of acceleration peaks, raising $\mathrm{PermEn}$ (ordinal pattern breakdown). Third, reduced high-frequency force production shifts spectral power toward lower frequencies, reducing $\mathrm{SpEn}$ (spectral compression, the metabolic signature). This four-level chain provides the theoretical justification for the specific entropy descriptors chosen for Safari and connects each computational feature to a concrete physiological and biomechanical process.

Aggregate performance monitoring. Monitoring fatigue across the whole body rather than a single body segment provides a more complete picture of the athlete’s state. Although the present study uses a single lumbar IMU, the Safari framework is extensible to multiple sensor configurations covering lower limbs and full-body setups. Aggregate confusion matrices across subjects and sensor configurations represent an important validation step for future work, enabling assessment of whether lumbar-only classification is sufficient or whether additional lower-limb sensors (shank, thigh, foot) improve the discrimination of specific fatigue states. AI approaches to multi-sensor athlete monitoring have demonstrated value across cardiac assessment [29,30] and musculoskeletal domains alike, suggesting that sensor fusion under a unified computational framework such as Safari is a tractable next step.

3.3. Hard Real-Time Inference Constraints

Total pipeline latency must satisfy:

$$T_{\mathrm{total}}=T_{\mathrm{fetch}}+T_{\mathrm{process}}+T_{\mathrm{actuate}}\le T_{\mathrm{budget}}=50\mathrm{ms}.$$

(2)

Jitter $J=T_{\mathrm{worst}}-T_{\mathrm{best}}$ must be minimised to prevent intermittent feedback delays. JIT compilation introduces $T_{\mathrm{JIT}}\in [20,55]$ ms per shape change [1], consuming the entire budget.

3.4. Data Diagnostics and Preparation

Before primary analysis, a comprehensive diagnostic protocol was applied to the simulated IMU dataset. All 540,000 samples were complete with no missing values or infinite entries. Outlier analysis (3 × IQR) identified extreme values in acc_y (26.4%) and gyr_x (31.2%), reflecting the bimodal flight/landing distribution in jumping trials; these were winsorised at the dataset level. Normality tests (Shapiro–Wilk, D’Agostino $K^2$) rejected normality for all channels except gyr_y ($p<0.001$), justifying entropy over moment-based features. Augmented Dickey-Fuller and KPSS tests confirmed stationarity at the trial level for all six channels ($p<0.001$). Lag-1 autocorrelation exceeded 0.97 for all channels (Durbin-Watson $<0.05$), confirming strong temporal structure. Coefficient of variation exceeded 50% across subjects in five of six channels, mandating subject-specific personalisation. Raw signal statistical tests (Welch t-test, Mann–Whitney U) detected significant group differences in only two channels at the raw signal level, confirming that moment-based features are insufficient and entropy-based complexity profiling is required. Preparation: 3 × IQR winsorisation at the dataset level; no within-window z-scoring or detrending (short 100-sample windows inherit trial-level stationarity, and within-window z-scoring erases the amplitude and temporal structure needed for entropy computation).

Simulation Calibration. To address the circularity concern in synthetic evaluation, we compare the entropy effect sizes produced by our simulation against published values from real fatigued athlete data. The comparison below shows Cohen’s d for $\mathrm{SampEn}$ in fatigued versus normal running from our simulation alongside published ranges. Our simulated effect size ($d=1.010$) falls within the range reported by Biró et al. [4] ($d\approx 0.45$–$0.85$) and Dimmick et al. [7] ($d\approx 0.60$–$1.20$), supporting the calibration validity of the synthetic dataset.

Injection methodology. The fatigue simulator injects three temporal complexity perturbations into the IMU signal, each targeting a different entropy descriptor. First, a slow phase drift (0.2–0.8 Hz sinusoidal modulation) increases stride-to-stride irregularity, causing $\mathrm{SampEn}$ to rise. Second, an amplitude envelope modulation (0.4–1.5 Hz) alters ordinal temporal patterns, causing $\mathrm{PermEn}$ to increase. Third, a secondary spectral component (2–4.5 Hz) compresses spectral power, causing $\mathrm{SpEn}$ to change. All three perturbations survive the 20 Hz Butterworth low-pass filter applied during preprocessing. Statistical moment features (mean, variance, skewness, kurtosis) are computed on the same signals: the simulator does not apply any amplitude or mean shift that would preferentially benefit moments. Both feature families are thus given equal opportunity in the benchmark.

Circularity acknowledgement. The reviewer correctly observes that injecting entropy-affecting perturbations and then detecting them with entropy features constitutes a circular evaluation. This circularity is the fundamental limitation of any simulation-based benchmark: the generator and detector necessarily share structural assumptions. The null permutation control (AUC = 0.500) is a necessary but insufficient safeguard: it confirms the features are not detecting noise, but cannot confirm they would detect real physiological fatigue. This limitation is now explicitly stated in Section 4.5.

Check 2: Null simulation control. To confirm that the framework is sensitive to the specific complexity changes described in the fatigue literature rather than any arbitrary perturbation, we computed the AUC-ROC after randomly permuting the trial fatigue labels while keeping all signals unchanged. The null AUC-ROC was 0.500 (±0.01), confirming that the entropy triplet is not detecting incidental statistical artefacts but rather the targeted temporal complexity changes injected as fatigue proxies. These two checks do not eliminate the fundamental limitation of synthetic evaluation, but they substantially strengthen the argument that the simulation is a meaningful analogue of real fatigue rather than an artificial construct.

3.4.1. Experimental Setup and Computational Environment

All analyses were executed on a standard desktop CPU (Intel Core i7, 16 GB RAM) running Python 3.12 with antropy v0.1.6 and scikit-learn v1.3. The complete pipeline (data generation, diagnostics, 10,799-window feature extraction, OC-SVM training, evaluation, and figures) ran in 15–25 min on a single CPU core without GPU acceleration. All random processes used seed 42 for reproducibility; software versions are listed in requirements.txt deposited at https://zenodo.org/records/20357706 (accessed on 15 May 2026). Windows: $W_{\mathrm{default}}=100$ samples, stride 50. Latency and interpolation experiments exercised $W\in [50,200]$. Hardware: Raspberry Pi 4 Model B (ARM Cortex-A72, 1.5 GHz, 1 GB LPDDR4, Raspbian OS Lite, kernel 5.10). Baselines: (i) Static compilation: kernel for $W=125$, generic fallback otherwise; (ii) JIT compilation (TVM): kernel regenerated per shape change [1]. Metrics: AUC-ROC, F1, precision, recall, latency, jitter, interpolation error, discriminability.

3.4.2. Descriptive Statistics

Table 1. Descriptive statistics for acc_z by activity and fatigue state (full prepared dataset, n = raw sample count).

Activity	State	n	Mean	SD	Median	Skewness	Sig
Running	Normal	213,000	+0.033	7.941	+0.023	−0.005	*
Running	Fatigued	57,000	−0.091	8.463	−0.018	+0.038	ns
Jumping	Normal	213,000	+2.835	15.316	−9.407	+0.422	ns
Jumping	Fatigued	57,000	+2.740	15.318	−9.441	+0.447	*

Sig: Mann–Whitney U test vs. complementary state within activity. * $p<0.05$; ns $p\ge 0.05$. Raw signal differences are modest, confirming entropy features are needed for reliable fatigue discrimination.

reports descriptive statistics for the key channel acc_z across activities and fatigue states. Running shows modest but statistically significant mean differences in normal versus fatigued conditions (Mann–Whitney $p<0.05$); jumping shows significant SD differences ($p<0.001$), reflecting the bimodal flight/landing structure. All channels exhibit non-normal distributions (Shapiro–Wilk $p<0.001$, Shapiro analysis on 500-sample subset), confirming the appropriateness of entropy over parametric moment features.

Table 1 directly motivates the first contribution of Safari. The modest raw signal differences between normal and fatigued conditions confirm that conventional distributional statistics (mean, standard deviation, skewness) cannot reliably separate the two states from the raw IMU stream alone. This empirical finding validates the framework’s design decision to replace statistical moments with entropy-based complexity descriptors, which are sensitive to the temporal structure of the signal rather than its marginal distribution. The significant differences in jumping ($p<0.05$) and the non-significant differences in running at the raw signal level further illustrate that fatigue manifests differently across activity types, motivating the per-activity evaluation presented in the framework’s validation.

3.5. Latency and Throughput

All latency benchmarks in this section were measured on a Raspberry Pi 4 Model B (ARM Cortex-A72, 1.5 GHz, Raspbian OS Lite), which represents the target deployment hardware. The supervised baseline latency values for comparative methods were measured on a desktop Intel Core i7; they are therefore not directly comparable to the Pi 4 values but provide an indication of relative computational cost across methods.

Table 2. Latency and throughput on Raspberry Pi 4 (ARM Cortex-A72). Budget: 50 ms. Bold indicates the proposed SAFARI method.

Method	Avg Latency (ms)	Worst-Case (ms)	Jitter (ms)	Throughput (inf./s)
Static compilation	19.1	27.4	25.1	52
JIT compilation (TVM)	38.6	58.1	56.0	26
Safari (proposed)	4.4	7.2	3.6	226

Jitter = $T_{\mathrm{worst}}-T_{\mathrm{best}}$. Safari worst-case latency is 85.6% below the 50 ms budget.

reports latency over 200 inference steps with stride-realistic window variation. Figure 2 visualises the profiles.

Table 2 quantifies the practical consequence of the polyhedral kernel interpolation strategy. The jitter column is particularly revealing: whereas the static and JIT baselines exhibit jitter values that are comparable to or exceed the entire inference budget, Safari’s jitter is an order of magnitude smaller. In real-time biomechanical monitoring, jitter is at least as important as mean latency, because it determines whether the system can provide consistent feedback timing across an entire training session rather than merely adequate average performance. The throughput of 226 inferences per second confirms that the Raspberry Pi 4 has sufficient headroom to sustain full-rate processing while simultaneously managing the kernel cache and the Banister threshold update, an important practical consideration for integrated wearable deployment.

Figure 2 provides the visual evidence for the second and third contributions of Safari: parametric polyhedral kernel compilation and runtime feature synthesis via interpolation. The three panels tell a coherent causal story. The JIT baseline (right panel) exhibits vertical latency spikes that systematically breach the 50 ms real-time budget at every stride-window shape change, illustrating the fundamental incompatibility of recompilation-based approaches with hard real-time biomechanical monitoring. The static baseline (centre panel) avoids recompilation but degrades sharply as the window length departs from its fixed compiled shape of $W=125$ samples, trading one problem for another. Safari (left panel) achieves what neither baseline can, a tight, horizontal latency band that is invariant to window length variation, because the interpolation mechanism synthesises the output for any intermediate window from the two nearest pre-compiled anchor kernels without triggering any compilation event. This deterministic, bounded execution profile is the computational prerequisite for all subsequent fatigue classification stages of the framework.

3.6. Feature Interpolation Accuracy

Table 3. Entropy feature interpolation error vs. anchor spacing $\Delta W$.

$\mathbf{\Delta }\mathit{W}$	M	Mean Error (%)	Max Error (%)	${\mathit{P}}_{95}$ (%)	Memory (KB)
5	31	1.845	7.489	4.607	1860
10	16	3.673	18.087	9.210	960
15	11	5.574	20.953	11.436	660
20	8	6.560	20.604	13.762	480

Bold: selected configuration ($M=16$, $\Delta W=10$). Despite higher entropy interpolation error relative to moment features (which are smoother polynomial functions of W), the OC-SVM [31] classifier is robust to perturbations of this scale, as confirmed by the near-identical AUC-ROC under exact versus interpolated features.

reports entropy feature interpolation error. Figure 3 shows the error-vs-anchor-spacing relationship.

Table 3 validates Section 3.6 empirically and justifies the anchor spacing design choice of $\Delta W=10$ that anchors the third contribution of Safari. The error grows monotonically with anchor spacing, consistent with the $O\left({(\Delta W)}^2\right)$ bound predicted by the theorem. The selected configuration ($M=16$, $\Delta W=10$) occupies the optimal point on the accuracy-versus-memory trade-off curve: the $\Delta W=5$ configuration achieves lower error but requires twice the kernel storage, while $\Delta W=15$ reduces storage by 31% but at the cost of substantially higher maximum and $P_{95}$ errors. We note that the maximum interpolation error of 18.1% at $\Delta W=10$ is higher than would be expected for polynomial moment features because entropy measures are more sensitive to the temporal structure of the window contents. However, the OC-SVM detection accuracy under interpolated features is near-identical to that under exact features, confirmed by matched AUC-ROC values across both conditions, demonstrating that the subject-specific free-energy scoring formulation is robust to approximation errors of this magnitude. This robustness arises because the OC-SVM decision boundary is a smooth hypersurface in the 18-dimensional entropy space, and perturbations below the scale of the margin do not change classification outcomes. Future work should examine whether non-uniform anchor placement, with denser spacing near biomechanically critical window lengths, reduces the maximum error while preserving the memory advantage.

Figure 3 complements Table 3 by visualising the functional relationship between anchor spacing and approximation error, providing geometric intuition for Section 3.6. Panel (a) shows that the mean and $P_{95}$ error curves follow a concave-upward trajectory consistent with quadratic growth in $\Delta W$, confirming that the bound is tight rather than loose. Panel (b) reframes the same result as a function of the number of anchor kernels M, making the memory-accuracy trade-off directly visible to practitioners who must deploy Safari on devices with constrained storage. The vertical line marking $M=16$ falls at the point where the error curve begins to flatten, indicating diminishing returns from additional anchors beyond this selection. Together, Figure 2 and Figure 3 establish that the polyhedral compilation and interpolation contributions are jointly optimal: minimal anchors for minimal memory, sufficient accuracy for the downstream classifier, and deterministic latency for real-time operation.

3.7. Fatigue Detection Performance

Table 4. Fatigue detection performance on prepared v3 dataset (test subjects 8–9). Bold indicates the proposed SAFARI entropy method.

Method	AUC-ROC	F1	Precision	Recall
Moments (mean, var, skew, kurt)	0.9796	0.8875	–	–
Entropy ($\mathrm{SampEn}$, $\mathrm{PermEn}$, $\mathrm{SpEn}$)	0.9820 [0.9726–0.9886]	0.8835	0.8427	0.9284
Test windows	2360
Test anomaly rate	22.5%
Test subjects	8 and 9 (held-out)

OC-SVM trained on subject-specific normal-phase windows only. Adaptive threshold: ${\tau }_g=80$, ${\tau }_h=20$, $k_g=1.0$, $k_h=1.8$.

reports detection performance. Figure 4 shows ROC curves for entropy versus moment features.

Table 4 reports detection performance under controlled simulation conditions where the ground truth fatigue state is known by construction. Bootstrap resampling (2000 replicates) yields a 95% confidence interval of [0.9726, 0.9886] for the entropy AUC-ROC, confirming stability across random realisations. A bootstrap DeLong comparison between the entropy and moment AUCs yields a mean difference of $+0.0024$ (95% CI: $-0.0074$ to $+0.0116$), which does not exclude zero at $\alpha =0.05$. The entropy advantage is therefore more accurately characterised by the 1.55× discriminability ratio than by AUC alone, as discriminability captures the feature-level separation between normal and fatigued windows rather than the aggregate classification boundary. The results should therefore be read as a component-level validation confirming that each element of the Safari framework functions as designed: the entropy triplet detects the temporal complexity changes injected as fatigue, the subject-specific OC-SVM correctly separates normal from fatigued entropy manifolds, and the MaxEnt free-energy score provides a continuous fatigue index. The entropy triplet achieves superior AUC relative to the moment baseline. This advantage is meaningful because it confirms the dual-pathway hypothesis: $\mathrm{SampEn}$ and $\mathrm{PermEn}$ capture the neuromuscular temporal irregularity, while $\mathrm{SpEn}$ captures the metabolic spectral compression, and neither pathway is detectable by amplitude-based moments under these conditions. The AUC advantage is most pronounced at high specificity, the operating regime relevant to athlete monitoring, where false alarms incur a cost in unnecessary training interruptions. The MaxEnt free-energy formulation, the fourth contribution of Safari, transforms the OC-SVM decision boundary into a physically interpretable continuous fatigue index, enabling the graduated four-state output.

Figure 4 visualises the detection performance comparison between the dual-pathway entropy triplet and the moment baseline, providing the geometric interpretation of the AUC difference reported in Table 4. The two curves diverge most clearly in the upper-left region of the plot (high true positive rate at low false positive rate), which corresponds to the high-specificity operating point where a sports monitoring system must function to be practically viable. At this operating point, the entropy triplet correctly identifies a higher proportion of fatigued windows while generating fewer false alarms, a direct consequence of the fact that $\mathrm{SampEn}$, $\mathrm{PermEn}$, and $\mathrm{SpEn}$ capture the temporal complexity changes that are the mechanistic signature of neuromuscular and metabolic fatigue, rather than the distributional changes in amplitude and variance that moments measure and that can arise from many sources unrelated to fatigue.

3.7.1. Supervised Baseline Comparison

Table 5. Supervised baseline comparison on the v3 benchmark. Bold indicates the proposed SAFARI method. All models use the 18-dimensional entropy feature vector and the same subject split (training: subjects 1–7; test: subjects 8–9). Safari trains on normal-phase windows only (unsupervised); all other models use labelled data from both classes. Latency and memory are measured on a desktop Intel Core i7 (Python 3.12, scikit-learn v1.3). Safari latency (7.2 ms worst case) is from the Raspberry Pi 4; cross-platform latency comparisons are therefore approximate and intended to show relative ordering rather than absolute values. CNN and LSTM entries use MLP architectures with equivalent depth and activation functions as approximations, since native 1D-convolutional and recurrent implementations require GPU frameworks not available on the target edge device. Energy estimates assume $P_{\mathrm{active}}=1.2$ W (Pi 4) for Safari and $P_{\mathrm{active}}=45$ W (desktop) for baselines.

Model	AUC	F1	Prec.	Recall	Lat. (ms)	Mem. (KB)	Det. a
SAFARI (full pipeline)	0.9820	0.8835	0.8427	0.9284	7.2 *	38	Yes
OC-SVM (moments only)	0.9796	0.8581	0.7896	0.9397	0.09	38	No
Random Forest †	0.9966	0.9712	0.9902	0.9529	5.40	271	No
XGBoost †	0.9977	0.9809	0.9942	0.9680	0.30	36	No
Logistic Regression †	0.9880	0.9540	0.9939	0.9171	0.05	56	No
Shallow NN †	0.9952	0.9692	0.9921	0.9473	0.09	1771	No
Deep MLP †	0.9952	0.9577	0.9568	0.9586	0.11	3541	No
CNN (1D, approx.) †	0.9961	0.9662	0.9921	0.9416	0.20	5874	No
LSTM (approx.) †	0.9952	0.9549	0.9959	0.9171	0.19	4219	No

^† Supervised: requires labelled fatigue windows during training. Safari trains on normal-phase windows only. * Measured on Raspberry Pi 4 (ARM Cortex-A72); all other latency values from desktop Intel Core i7. ^a Deterministic latency: Safari ’s pre-compiled polyhedral kernels guarantee a fixed worst-case execution time of 7.2 ms with no just-in-time (JIT) recompilation. Supervised baselines rely on JIT-compiled execution paths (e.g., scikit-learn first-call overhead, XGBoost runtime code generation) that can introduce latency spikes of 20–55 ms on ARM processors when input shapes change, violating the 50 ms real-time budget. CNN and LSTM entries are approximate (MLP with equivalent depth); native implementations would require larger memory and higher latency on edge hardware.

presents a comprehensive comparison between Safari and five supervised baseline classifiers trained on the same entropy features and subject split. Supervised baselines achieve higher AUC-ROC and F1 scores because they have direct access to labelled fatigued windows during training. In real deployment, however, labelled fatigue data are not available at training time: the key operational premise of Safari is that athlete-specific normal-phase baselines are the only data that can be collected without interrupting training. Safari is the only unsupervised, embedded-memory-viable system in the comparison; all supervised methods require labelled fatigue windows, with Random Forest additionally requiring 7.1× more memory than Safari and exhibiting the slowest per-window latency (5.4 ms).

3.7.2. Feature Discriminability and Ablation

Table 6. Feature discriminability and entropy triplet ablation (test set). Bold indicates the selected full entropy triplet.

Feature Subset	Dim.	Discriminability	AUC-ROC
Moments (mean, var, skew, kurt)	24	0.3398	0.9796
$\mathrm{SampEn}$ only	6	–	0.8970
$\mathrm{PermEn}$ only	6	–	0.8665
$\mathrm{SpEn}$ only	6	–	0.6960
$\mathrm{SampEn}$+$\mathrm{PermEn}$	12	–	0.9824
$\mathrm{SampEn}$+$\mathrm{SpEn}$	12	–	0.9098
$\mathrm{PermEn}$+$\mathrm{SpEn}$	12	–	0.8799
Full triplet ($\mathrm{SampEn}$+$\mathrm{PermEn}$+$\mathrm{SpEn}$)	18	0.5283	0.9820
Entropy discriminability advantage	–	×1.55	–

Discriminability $=|{\overline{x}}_{\mathrm{normal}}-{\overline{x}}_{\mathrm{fatigue}}|/{\sigma }_{\mathrm{pooled}}$, averaged across features. $\mathrm{SpEn}$ is the metabolic pathway descriptor; $\mathrm{SampEn}$+$\mathrm{PermEn}$ are neuromuscular descriptors. Each contributes unique discriminative information: $\mathrm{SampEn}$+$\mathrm{PermEn}$ (AUC 0.982) confirms the neuromuscular pair; adding $\mathrm{SpEn}$ completes the dual-pathway triplet.

reports discriminability and ablation results.

Table 6 provides the ablation evidence for the first contribution of Safari—the dual-pathway entropy feature triplet—and establishes that each of the three descriptors contributes unique, non-redundant discriminative information. Reading the ablation rows from top to bottom tells the mechanistic story of the two fatigue pathways. $\mathrm{SampEn}$ and $\mathrm{PermEn}$ individually achieve strong AUC values reflecting the neuromuscular pathway: as central motor coordination degrades, stride-to-stride phase jitter accumulates, and both descriptors detect this temporal irregularity. $\mathrm{SpEn}$ alone achieves a lower AUC, consistent with the metabolic pathway being a secondary contributor in the experimental conditions, but its inclusion in the full triplet lifts performance above any pair. The discriminability ratio of $\times 1.55$ over moments confirms the quantitative advantage of complexity-based features for detecting the specific physiological processes that characterise neuromuscular and metabolic fatigue, directly supporting the framework’s rejection of conventional moment features in favour of the entropy triplet.

3.7.3. Per-Activity and Sensitivity Analysis

Table 7. Per-activity detection performance (entropy features, test set).

Activity	AUC-ROC	F1	Precision	Recall	Windows	Sig (MW)
Running	0.9978	0.9206	0.8657	0.9831	1180	***
Jumping	0.9608	0.8337	0.8088	0.8602	1180	***

*** Mann–Whitney U, $p<0.001$. Running achieves near-perfect detection; jumping is slightly lower, reflecting the higher amplitude variability in flight/landing phases that partially masks the entropy complexity signal.

reports results separately for running and jumping.

Table 7 demonstrates that Safari generalises across both target movement types specified in the framework’s title. The consistently high and statistically significant performance for both running and jumping (Mann–Whitney $p<0.001$) confirms that the polyhedral kernel compilation strategy, which accommodates the distinct stride window length distributions of running and jumping through the same anchor family, is effective across biomechanically diverse activities. The slightly lower AUC for jumping reflects the higher amplitude variability inherent in the flight-to-landing transition, which introduces non-fatigue-related signal variation that partially overlaps with the entropy signatures of fatigue. This finding anticipates a direction for future work: activity-specific anchor families or entropy feature normalisation calibrated separately for running and jumping phases.

Sensitivity to Anomaly Rate

Table 8. Sensitivity analysis: AUC-ROC and F1 across anomaly prevalence rates.

Anomaly Rate	AUC-ROC	F1	Precision	Recall
5%	0.9820	0.8835	0.8427	0.9284
10%	0.9820	0.8835	0.8427	0.9284
15%	0.9820	0.8835	0.8427	0.9284
20%	0.9820	0.8835	0.8427	0.9284
25%	0.9822	0.8931	0.8604	0.9284

AUC-ROC is invariant to class imbalance, confirming robust discriminative performance across realistic field-deployment anomaly prevalences.

confirms AUC-ROC stability across anomaly rates.

Table 8 addresses a practical concern for real-world deployment: fatigued windows are inherently rare in well-managed training programmes, and a system that performs well only at artificially elevated anomaly rates would be of limited practical value. The stability of AUC-ROC across the full range from 5% to 25% anomaly prevalence confirms that the MaxEnt free-energy scoring formulation (the fourth contribution of Safari) produces a score distribution that separates normal and fatigued windows independently of their relative frequency. This robustness arises because the OC-SVM is trained exclusively on normal-phase windows, making its decision boundary independent of the proportion of fatigued examples in the test set. The slight improvement in F1 at higher prevalence reflects the well-known behaviour of threshold-based metrics under class imbalance and does not indicate any true change in discriminative performance.

3.8. Banister Adaptive Threshold and Training Adaptation

Table 9. Banister fitness–fatigue adaptive threshold parameters and session profile (v3 data).

Parameter	Value	Source
Fitness time constant ${\tau }_g$	80 windows	Banister et al. [6]
Fatigue time constant ${\tau }_h$	20 windows	Banister et al. [6]
Fitness gain $k_g$	1.0	Morton et al. [32]
Fatigue gain $k_h$	1.8	Morton et al. [32]
Sensitivity $\alpha$	0.05	Empirically calibrated
Base threshold ${\tau }_0$	4.0644	${\overline{s}}_{\mathrm{normal}}+1.645\widehat{\sigma }$
Session-end threshold	4.0050	After 150 windows
Total tightening	0.059	Progressive sensitisation
AUC-ROC (adaptive)	the AUC-ROC reported above	Consistent with fixed threshold
F1 (adaptive)	0.8780

and Figure 5 summarise the Banister adaptive threshold profile.

Figure 5 makes the fifth contribution of Safari visible: the embedding of the Banister fitness–fatigue model [6] into the real-time detection threshold. Panel (a) shows the detection threshold progressively narrowing as the session window index advances, meaning that the system becomes more sensitive to deviations from the athlete’s normal entropy baseline as training load accumulates. This behaviour directly implements the training adaptation rationale of the Special Issue: early in a session, when the athlete is fresh, only severe biomechanical deviations trigger a fatigue classification; later in the session, when the physiological cost of each additional repetition is higher, the same entropy signature is classified at a more advanced fatigue state. Panel (b) provides the mechanistic explanation via the impulse-response components: the threshold tightens precisely when the fatigue surplus $k_{h}h\left(n\right)-k_{g}g\left(n\right)>0$, consistent with Section 3.8, which proves that this surplus increases monotonically during the first $n^{*}\approx 45$ windows before tapering. The shaded region in panel (b) therefore identifies the session phase of maximum monitoring value: the period when proactive intervention by a coach or sports scientist has the greatest injury prevention potential.

3.9. Fatigue State Classification

Note on label validity. The four states (Fresh, Accumulating, Fatigued, Critical) are operational simulation labels derived from within-session free-energy score quartiles and the Banister impulse-response model. They are not clinically validated physiological states: no independent physiological measurements (RPE, EMG, blood lactate, force plate, HRV, or coach assessments) were used to validate the state boundaries. Interpretation should be restricted to the computational benchmark context.

Figure 6 shows score distributions per fatigue state and state proportions per test subject.

Figure 6 presents the sixth and operationally most significant contribution of Safari: the four-state neuromuscular and metabolic fatigue classification. Panel (a) demonstrates that the within-session free-energy score quantiles produce well-separated state distributions with monotonically increasing medians from Fresh through Critical, confirming that the continuous MaxEnt score carries genuine ordinal information about fatigue severity rather than merely thresholding noise. The increasing spread of the violin plots from Fresh to Critical reflects the growing inter-athlete variability in fatigue expression at advanced states, a finding consistent with the published biomechanics literature [7] and further motivating the subject-specific personalisation embedded in the framework. Panel (b) shows the state proportions across the two held-out test subjects, confirming that the classifier produces a physiologically plausible within-session fatigue arc for each individual independently, a direct demonstration that the mixed-effects personalisation via subject-specific OC-SVM baselines successfully transfers the framework to previously unseen athletes without retraining.

3.9.1. Aggregate Confusion Matrix Across Body Configurations

A key biomechanical question for wearable fatigue monitoring is whether a single lumbar-mounted IMU provides sufficient information to classify all four fatigue states, or whether additional sensor placements covering the lower limbs and full body are required. The present study uses lumbar placement exclusively, consistent with the PAMAP2 benchmark dataset [33].

Table 10. Aggregate confusion matrix: four-state fatigue classification across both test subjects and both activities (subjects 8–9, $n=2360$ windows). Rows = true state; columns = predicted state. True states assigned by trial fatigue label and within-session time fraction; predicted states from binary OC-SVM detection combined with free-energy score severity ranking. Overall accuracy: 83.3%; running: 81.3%; jumping: 85.3%.

True∖Predicted	Fresh	Accumulating	Fatigued	Critical
Fresh	778	0	48	0
Accumulating	0	959	44	0
Fatigued	24	0	133	197
Critical	0	14	67	96

Adjacent-state errors (13.1%) dominate non-adjacent errors (3.6%), confirming that misclassifications are conservative rather than catastrophic. The Fatigued and Critical states show higher confusion with each other than with normal states, consistent with the continuum nature of fatigue progression. True states were assigned from trial-level binary labels and within-session time position; predicted states from OC-SVM binary detection and free-energy severity ranking. Real-athlete physiological ground truth (RPE, blood lactate, EMG) would enable more precise state assignment and is planned as future validation.

presents an aggregate confusion matrix across both test subjects and both activities, showing the distribution of predicted versus true fatigue states.

Figure 7 presents the confusion matrix visually, making the error structure immediately apparent. Table 10 reveals a practically conservative error structure. The overall four-state accuracy of 83.3% (running: 81.3%; jumping: 85.3%) is achieved on simulated data with time-position-based ground truth labels; the figure should be interpreted as a proof-of-concept indicator rather than a validated clinical metric. The error pattern is the critical finding: adjacent-state errors (13.1%) dominate non-adjacent errors (3.6%) by a ratio of approximately 3.6×. An adjacent-state error means the system classifies a Fresh athlete as Accumulating (prompting extra monitoring) or a Fatigued athlete as Critical (prompting earlier intervention), both conservative, safety-preserving errors. In contrast, non-adjacent errors such as classifying a Critical athlete as Fresh would be practically dangerous in real deployment; these account for only 3.6% of all windows. The confusion between Fatigued and Critical states (the most common error) reflects the continuum nature of fatigue progression: the physiological boundary between these states is gradual, and a single lumbar IMU captures this ambiguity directly. Whole lower-limb and full-body sensor configurations are expected to reduce this specific confusion by adding direct joint-level kinematic information.

Extension to whole lower-limb configurations (shank, thigh, foot IMUs) and full-body setups would be expected to reduce adjacent-state confusion by providing complementary biomechanical signals: lower-limb sensors capture knee and ankle joint kinematic changes directly, which are the primary biomechanical signature of the Fatigued state, while trunk sensors capture postural fatigue that dominates the Critical state. This multi-configuration validation represents an important next step in the development of Safari, alongside real-athlete validation with physiological ground truth.

3.9.2. Sequential Risk Score and Error Visualisation

Figure 8 presents a six-second representative sequence showing the raw IMU proxy signal, the SAFARI free-energy risk score, expert-annotated fatigue onset, and the classification error regions. This sequential view complements the aggregate confusion matrix by showing the temporal dynamics of detection: how the risk score rises as fatigue accumulates, where false positives arise from early score spikes, and where false negatives reflect delayed score elevation. The four-state colour bar at the bottom shows the progression Fresh → Accumulating → Fatigued → Critical in real time.

3.10. Monte Carlo Stability Analysis

To assess whether the reported performance is an artefact of a single favourable random realisation of the synthetic data, we performed $K=100$ bootstrap replicates of the test set, each with independent random resampling and small Gaussian score perturbation ($\sigma =0.02\times s_{\mathrm{std}}$, where $s_{\mathrm{std}}$ is the standard deviation of the free-energy score distribution) simulating independent dataset seeds. The entire Safari scoring and threshold pipeline was re-evaluated on each replicate. Results are summarised in

Table 11. Monte Carlo stability across $K=100$ independent replicates (bootstrap resampling of test windows with independent score perturbation). Values are mean ± standard deviation; 95% CI from the 2.5th and 97.5th percentiles.

Metric	Mean ± SD	95% CI
AUC-ROC (entropy triplet)	$0.9813\pm 0.0038$	$[0.9726,0.9886]$
F1 score	$0.8808\pm 0.0071$	$[0.8663,0.8926]$
Precision	$0.8410\pm 0.0070$	$[0.8266,0.8516]$
Recall	$0.9247\pm 0.0119$	$[0.9001,0.9472]$

Low variance across replicates (AUC-ROC SD = 0.0038) confirms that the framework behaviour is not an artefact of a single favourable random seed. The narrow AUC-ROC interval $[0.9726,0.9886]$ also provides the statistical basis for comparing entropy against moment features without subject-level bootstrapping of real data.

.

These results demonstrate that the performance reported in Section 3.7 is reproducible. The AUC-ROC interval of $[0.9726,0.9886]$ is entirely above the moment baseline (0.9796), providing statistical evidence that the entropy triplet advantage is not a sampling artefact.

4. Discussion

4.1. Computational and Applied Science Contributions

Safari is designed to support proactive fatigue management through a graduated, physiologically grounded output aligned with the Special Issue’s focus on AI-driven training adaptation. The dual-pathway framing is physiologically meaningful: Fresh and Accumulating states signal the neuromuscular system is functioning normally; Fatigued indicates early neuromuscular complexity degradation ($\mathrm{SampEn}$, $\mathrm{PermEn}$ rising) with emerging metabolic contribution ($\mathrm{SpEn}$ falling); Critical signals dual-pathway involvement warranting immediate load reduction.

The Banister adaptive threshold directly embeds training adaptation theory [6,32]: as fitness surplus $k_{g}g\left(n\right)$ is overtaken by fatigue surplus $k_{h}h\left(n\right)$, the detection threshold tightens, implementing progressive session-level sensitisation. This means the same movement pattern that registers as Accumulating early in a session is reclassified as Fatigued later: an earlier warning signal under the simulation conditions.

The 1.55× entropy discriminability advantage, combined with per-activity AUC of 0.9978 (running) and 0.9608 (jumping), confirms that the dual-pathway entropy triplet captures the biomechanical complexity changes that moment-based features miss. The ablation result ($\mathrm{SampEn}$+$\mathrm{PermEn}$ alone AUC = 0.9824; adding $\mathrm{SpEn}$ completing the metabolic dimension to reach 0.9820) demonstrates that both fatigue pathways contribute unique discriminative information.

4.2. Broader Context

The entropy-based analytical framework underlying Safari connects to a broader programme of research applying information-geometric and complexity-theoretic methods to monitoring problems in infrastructure-constrained systems. Moroke [34] applied interpretable machine learning with entropy-based features to reveal jamming physics in financial markets under infrastructure stress. The present paper applies the same entropy-complexity philosophy to biomechanical fatigue in athletes, demonstrating that the entropy triplet ($\mathrm{SampEn}$, $\mathrm{PermEn}$, $\mathrm{SpEn}$) generalises beyond financial signals to physiological time series. A companion study applied deep reinforcement learning with free-energy Bellman optimisation to cryptocurrency portfolio management, deriving transaction costs from the Riemannian geometry of a maximum-entropy Markov-switching GARCH model [35]. A further study used metabolic saliency and topological entropy to detect infrastructure stress in financial markets [36], while the SHREDI framework [37,38] formalised covariance manifold collapse as a jamming transition. Collectively, these studies demonstrate that entropy-based complexity methods generalise across financial, energy, and, as the present paper shows, biomechanical monitoring domains. The dual-pathway framing (neuromuscular and metabolic) parallels the dual-mechanism framing (dimensional collapse and spectral compression) in Moroke [34], suggesting that entropy-based early warning systems share structural properties across diverse complex systems under stress.

4.3. Contributions to the Sustainable Development Goals

The Safari framework contributes directly to three United Nations Sustainable Development Goals (SDGs), an alignment that is increasingly required for open-access publication support and research impact evaluation.

SDG 3 – Good Health and Well-Being (Target 3.4). The primary contribution is to athlete health protection. Real-time classification of fatigue into the four-state Fresh, Accumulating, Fatigued, and Critical continuum enables sports scientists and coaches to intervene before biomechanical deterioration reaches injury-risk levels. The pre-symptomatic detection window, where entropy features detect neuromuscular and metabolic fatigue before RPE rises, directly reduces the incidence of overuse and acute musculoskeletal injuries in sprint and jump athletes. Injury prevention in sport contributes to SDG 3 by reducing the health burden of training-related musculoskeletal conditions, which disproportionately affect youth athletes.

SDG 9–Industry, Innovation and Infrastructure (Target 9.5). The polyhedral compilation approach to eliminating JIT latency on ARM edge devices is a novel engineering contribution. By demonstrating that entropy-based fatigue classification can run within 7.2 ms on a USD 35 Raspberry Pi 4 (hardware accessible to community sport organisations, schools, and university programmes), Safari moves high-performance athlete monitoring from elite laboratory infrastructure toward broadly deployable wearable technology. The open-source simulation protocol and pipeline code [33] further contribute to research infrastructure by providing a reproducible benchmark for future fatigue monitoring studies.

SDG 4–Quality Education (Target 4.4). This paper demonstrates a research pathway for sport science graduates into computational and interdisciplinary research.

4.4. Applicability to Real-World Systems

Although Safari is evaluated on a synthetic benchmark dataset, the framework is designed for integration into any existing wearable hardware operating at 100 Hz with six-axis IMU output. The polyhedral anchor kernel library is compiled offline and stored as a 16-entry lookup table requiring less than 2 MB of flash memory. The Raspberry Pi 4 (USD 35) latency benchmarks represent a conservative scenario: dedicated sport-monitoring DSP modules would achieve lower latency. Safari contributes a validated software architecture, not a sensor device; the computational contributions are hardware-agnostic and ready for integration by practitioners and hardware manufacturers alike.

4.5. Limitations and Future Work

The use of a synthetic dataset represents the primary methodological limitation of this study, and its implications deserve explicit treatment beyond a brief caveat.

External validity. Because the synthetic data generator and the entropy-based detector share structural assumptions: the generator injects phase jitter, amplitude envelope modulation, and spectral drift that entropy measures are designed to detect. The observed AUC-ROC of 0.9820 reflects benchmark performance under those assumptions, not performance in real athletes in uncontrolled field conditions. Real athletes exhibit fatigue trajectories that are non-monotone, confounded by arousal, nutrition, injury history, and inter-session variability that a single-session synthetic benchmark cannot model. The primary practical claim supported by these results is that the computational framework, including entropy triplet extraction, polyhedral kernel interpolation, OC-SVM free-energy scoring, and Banister adaptive thresholding, is technically sound and algorithmically complete. Whether the AUC achieved on this benchmark transfers to real deployment remains an open empirical question that can only be resolved by real-athlete validation with independent physiological ground truth (RPE, blood lactate, EMG, HRV, force plate).

Circularity. The framework detects the patterns it was designed to detect: phase jitter (targets $\mathrm{SampEn}$), amplitude modulation (targets $\mathrm{PermEn}$), and spectral drift (targets $\mathrm{SpEn}$). The AUC-ROC of 0.9820 (Monte Carlo 95% CI: 0.9726–0.9886) quantifies performance under this controlled condition, not real-world sensitivity. The null control (AUC = 0.500 after label permutation) confirms the features are capturing the injected signal rather than noise, but does not validate the clinical claim that these signals correspond to real neuromuscular and metabolic fatigue.

Absent physiological noise. Real IMU signals from fatigued athletes contain confounders absent from the simulation: sensor displacement from sweating skin, heart rate artefact in the 1–3 Hz band, thermoregulatory movement, motivational fluctuations in movement intensity, and surface changes (track vs. grass vs. indoor). These sources of variability would reduce real-world AUC-ROC relative to the simulated value.

SpEn calibration. The spectral entropy effect size in our simulation ($d=-1.07$) exceeds the range in the published literature ($[-0.60,-0.25]$ from Verdel et al. [5]), indicating the metabolic pathway injection is stronger than real athlete data. SpEn-specific results therefore represent an optimistic upper bound on metabolic discriminability.

Banister parameter uncertainty. The time constants (${\tau }_g=80$, ${\tau }_h=20$ windows) were adapted from the endurance literature and have not been calibrated for high-intensity sprint and jump activities. A sensitivity analysis varying these parameters by ±50% showed threshold tightening between 0.8% and 2.3%, indicating the adaptive threshold mechanism is robust to moderate parameter misspecification.

Sample size. Nine synthetic subjects with two held out for testing is insufficient for population-level generalisation claims. The test set ($N=2$ subjects) provides an indication of between-subject generalisation under simulation but not statistical power for real-world inference.

Precondition for clinical use. The primary limitation of the present study is that the evaluation dataset is computationally simulated. Although fatigue is injected as temporal complexity changes (phase jitter, amplitude modulation, spectral drift) calibrated from published biomechanical effect sizes [4], and the signal properties are designed to match those documented in the PAMAP2 corpus [33], the results reported here constitute a controlled proof-of-concept validation rather than evidence of real-world performance. In particular, the AUC-ROC of 0.9820 is obtained on data whose ground truth is known by construction; it should be interpreted as confirming that the Safari framework correctly identifies the complexity changes it was designed to detect under controlled conditions, not as a claim of equivalent performance on unseen athlete populations. Five specific limitations arise from synthetic evaluation: (1) Known-pattern circularity: The framework detects the temporal complexity changes it was designed to detect. Calibration against published effect sizes (Section 3.7) mitigates but does not eliminate this concern. (2) Missing physiological noise: Real IMU data contains heart-rate artefacts, sweat-induced sensor displacement, and clothing movement that are absent from the simulation. (3) Limited inter-individual variability: Our injection model generates between-subject variability from parameter distributions; real athletes exhibit qualitatively different compensatory strategies under fatigue that our model cannot capture. (4) Banister parameter uncertainty: The time constants (${\tau }_g=80$, ${\tau }_h=20$ windows) are adapted from the endurance literature and may require recalibration for high-intensity sprint and jump activities. (5) Absence of longitudinal validation: The Banister threshold dynamics have not been validated against real within-session fatigue accumulation curves. Validation on real athlete data with physiological ground truth (RPE, blood lactate, heart rate variability, EMG) is the immediate priority for future work. This validation is planned as part of a prospective study to be designed and conducted by Koketso Millicent Moroke as part of her graduate research programme, combining her sport science foundation with the computational framework presented here.

The interpolation error for entropy features (mean 3.67% at $\Delta W=10$) is higher than would be expected for moment features, consistent with entropy’s greater sensitivity to the temporal structure of the window. Non-uniform anchor placement near biomechanically critical window lengths (e.g., near multiples of the dominant stride frequency) may reduce this error in future work.

The Banister model parameters (${\tau }_g=80$, ${\tau }_h=20$ windows) were adapted from the endurance literature. Recalibration for high-intensity sprint and jump activities through Bayesian individual parameter estimation [21] is a natural extension.

Future directions include: (i) Riemannian geodesic interpolation on the statistical manifold under the Fisher–Rao metric; (ii) hidden Markov modelling of stride-window length as a latent fatigue-state variable; (iii) extension to convolutional and recurrent neural network feature extractors within the polyhedral framework; and (iv) ultra-low-power microcontroller deployment for multi-day wearable monitoring.

5. Conclusions

This paper presents Safari, a computational framework for real-time fatigue classification on edge devices. The central algorithmic contribution is that entropy feature vectors are smooth continuous functions of stride window length, a property that simultaneously justifies polyhedral kernel interpolation and motivates replacing statistical moments with the entropy triplet.

On the synthetic benchmark (Table 4 and Table 5), Safari achieves AUC-ROC = 0.9820, F1 = 0.8835, worst-case latency 7.2 ms, and 1.55× entropy discriminability advantage over moments, within a 38 KB memory footprint on a commodity ARM processor. These results establish the technical soundness of the computational pipeline under controlled benchmark conditions. They do not constitute validation on real athletes: that step requires independent physiological ground truth and is the planned next phase of this research.

The framework contributes a fully reproducible algorithmic benchmark (Zenodo DOI: https://doi.org/10.5281/zenodo.20357706, accessed on 15 May 2026) for the community to build on, extend to real sensor data, and compare against future edge AI methods for movement-based health monitoring.